Introduction The circuit of Fig. 1 is required to be modeled using a state - space representation, where 2 states will be used, based on the number of the energy - storing elements of the circuit, the capacitor and the inductor. u(t) + ΙΩ www 13 F 5 Ω it (t) www vc(t) 1 H Figure 1: LCR circuit The input signal to the circuit is the voltage u(t) in Volts and the output signal is the voltage across the capacitor, vc(t). Questions 1. Choice of system states: Choose appropriate signals for the 2 states of the system. x₁(t) = i₁(t) x₂ (t) =

5. State transition matrix: (t), which is defined as, Calculate analytically the state transition matrix (t) = et = L¯¹{(sI – A)¯¹} Show that the answer is the following, 1 e-4t cos(√2t) - e-4 sin(√2 t) 1 e -4t √2 (t) = et -3 1 -4t sin (√2 t) e COS -4t cos (√2t) + - e sin(√2 t) 2-4t sin(√2 t)| Calculate the following: (SI - A)-1= Use the completion - in - the-square technique (CASE 3) to calculate the inverse Laplace: L¯¹{(SI - A)¯¹} =

A single-core cable working on 66 kV has a conductor diameter of 2 cm and the sheath of inside diameter is 10 cm. If two metallic intersheaths of diameters 5 cm, 8 cm respectively are used for grading the cable.. If the maximum electric stress is the same for each layers. 1- Find the voltage of each metallic intersheaths. 2- Find the thickness of each layers.